I'm a DM for the first time and I'm running LMoP with a group of 5 very experienced players.

I have difficulties in handling situations in which the players try to intimidate the enemy roleplaying the intimidation with "scenic effects".
For example, suppose they meet with a random group of goblins which is looking for them.

They usually start casting lots of spectacular cantrips or 1st level spells; like the flame becomes blue, a thunder comes from clear sky, a deep voice from nowhere pretends to be the Maglubiyet the God of Goblins saying to beg on their knees, a little earthquake moves the terrain, and so on. So, since the group of goblins is supposed to be not intelligent, I always feel like the goblins should become submissive, docile, compliant, subjugated, and so they refuse to fight, or they run, or they get to the floor praying, and so on. If the goblins ignore this kind of intimidation, it doesn't feel right to me. And I don't really like the feeling when it seems that the players control my monsters.

What is the best way to handle these situations?

Treat it as a contest

The players are effectively trying to use a level 0 cantrip to produce the same result as a higher level crowd control spell like Fear or Mass Suggestion. Obviously, this is game breaking.

We had an arcane trickster rogue in our group who tried this kind of thing a lot. The way our DM handled it was to treat it as a contest. The caster is trying to deceive the goblins into thinking that they should not attack the party using the effect of a spell.

The caster casts the cantrip and then the makes a Deception or Persuasion or Intimdation skill check against the goblins' Insight checks.

You could have the goblins roll a single combined check or individually. If you do the latter, any goblins that are not convinced by the cantrip's effect can rally their comrades by saying something like "It's a trick! That's not what Maglubiyet sounds like!" (or teaches, whatever).

Rolling like this gives the goblin pack a high chance of resisting the cantrip's intended effect.

Alternatively, you can split the difference and have some of the goblins sit in one place for a round, quivering in their boots, while the unaffected ones charge forward. The scared goblins buck up the following round and join the braver ones.

Doing it this way allows you to keep the solution "within the game." You don't have to have a meta discussion with your player(s) to tell them that you won't let them do something. That's never fun.

As Kyle points out in his answer: At 10 intelligence and 8 wisdom, Goblins aren't actually "dumb." They're literally of average intelligence with a tinge of obliviousness, and in a world where magic is quite commonplace.

They're probably familiar with many illusion cantrips and Thaumatury, Minor Illusion, and Prestidigitation have a verbal and/or somatic component as part of their casting. If the spellcaster is visible, they'll see or hear him/her casting the spell first. Mighty suspicious, that.

Also, don't forget that, if a character is going to fake the god of the goblins speaking from on high, s/he will need to know the goblin language first!

Goblins are not actually dumb, unless the ones in Lost Mines of Phandelver have significantly different stats from normal. By default they have an intelligence score of 10, the same as an average person, and an 8 wisdom, which is only slightly worse. They are perfectly capable of recognizing the gestures and chanting of spellcasting and determining that it is a trick designed to intimidate them. They live in a world where illusions and magic are commonplace, so they are far more likely to assume that they are being tricked than that their god is directly speaking with them.

By default I personally would not allow any such attempt to work at all unless the spellcaster both concealed themselves from view so that the casting was not obvious and made a successful Deception check, and even then I would probably only give advantage on the Intimidation roll.

They are trying to intimidate, so they make an Charisma (Intimidation) check

The spells do not do anything other than what they say; however, if you feel that the spells etc. would be advantageous then give them advantage on the check.

Note that this is giving the spells more power than they have, but rewarding good play may justify such house-ruling.

Magic trickery grants advantage on checks

The rules for advantage state:

The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.

The circumstances in this case refer to their use of magic to improve their chance of success. Magic itself do not allow the intimidation to succeed automatically.

As Kyle Doyle states goblin's aren't actually dumb and should follow the same social rules as any other NPC.

Often at my table players try to use low level illusion spells to replicate the effects of a higher one. As a resolution to the multiple disputes about this we ruled that any situational assistance (magical or otherwise) simply grants advantage on an apposed check.

The goblins will roll insight to see through the attempt to influence them. The players roll either intimidation, deception or performance depending on the style of what they are trying to do.

There are many ways you can handle this. Allowing this all the time is overpowered and will likely get boring, but equally disallowing it all the time is boring and stifles creative play.

You have to find a balance, and that balance depends on the party. Personally, some of the most fun D&D I've had is when we make a half baked plan and manage to convince the bad guys that we are friends or similar, but other people prefer more straight combat encounters and want to get straight to the combat. You have to read the party - if they are all getting really into the strategy, then make more allowances for them to get it to work. Conversely, if 3 people are ready to fight and 1 person is trying to make this work then consider making it a simple intimidation check to make a couple of goblins run away while the rest attack.

That said, it seems that your party is into this sort of thing, so here's a couple of things to consider when coming up with (cool and fun) ways to deal with these situations.

Goblins aren't stupid

While it can be fun to play goblins are really stupid, you don't have to, and RAW they are of average intelligence. Goblins can figure out this isn't a miracle (or at least, not a miracle of their god), and then it's not going to be very effective. Or you can take the best of both worlds - the general goblins don't realise, but the goblin shaman realises what you are doing and convinces the goblins to attack (or starts putting up a second show to convince the goblins to attack).

Do they know enough about goblin culture?

Perhaps the goblins are submissive in the face of their deity. Or it could be that only goblins are 'holy' enough to see their deity, and the party (who are presumably not goblins) will now face fanatical goblins who will do anything to see them dead.

Since you're running a module with experienced players, they are more likely to know about things in the world already, but their characters don't necessarily know what the players know. Do they know who the god of goblins is? What they look like? How they act? You could make them make a religion check to figure out if they know enough and to avoid a faux pas ("I am the god of wrath! you should show mercy to these adventurers" would probably raise a few eyebrows). A deception check may be necessary to keep your lies straight.

There's also the consideration of language. The god of goblins will likely speak goblin - if the party doesn't know goblin, it will be a hard sell to say why the god of goblins is speaking common.

"Yes, but..." or "No, but..."

This is something seen in a few other RPGs, but just because it's not written into D&D doesn't mean you can't use it. Succeeding at a cost or failing with a benefit is a way to reward the creative play without making it overpowered.

They succeed at their religion check, but you can't remember any suitable goblin deity except XXX, a god who demands constant sacrifice. Now the adventurers are faced with a choice - is avoiding this fight worth increasing goblin raids on neighbouring villages to appease their god?

Alternatively, you fail a deception check, but the goblins are distracted by the show long enough that that you can sneak away (or at least get a head start), or sneak into better positions to get the drop on them.