Learn Dutch: Laten we het spandoek vasthouden om de tekst te tonen.

Laten we means "Let's" or "Let us". It is always paired with another verb to show what you are suggesting to do. In Dutch, this second verb (vasthouden) must be in its full dictionary form (the infinitive) and is pushed to the very end of its clause, right before the om ... te part begins.

I remember that vasthouden is a separable verb. Why aren't vast and houden split up here?

You are right that vasthouden is separable, meaning you would normally say Ik houd het spandoek vast (I hold the banner). However, separable verbs only split when they are the main conjugated verb in the sentence. Because vasthouden is acting as the infinitive pushed to the end by laten we, the prefix and the verb stay glued together.

Why do we use om ... te here? Can't we just say te tonen?

The structure om ... te translates to "in order to". While English often drops "in order" and just says "to show", Dutch requires the full om ... te structure when expressing a goal or purpose. The word om starts the phrase, and te goes immediately before the infinitive verb (tonen) at the end.

Could I just use houden instead of vasthouden?