Snare spell: when do creatures get an Intelligence Check to spot it?

There are two ways to do the check, depending on what the creature is doing.

1. A creature actively searches for traps.

If a creature is actively searching the room or whatever location you have hidden the snare spell, the DC for the check to find the snare trap is the spell save DC of the caster, and they make an active Investigation check (roll the d20).

Note, in combat, searching for traps requires taking the search action:

When you take the Search action, you devote your attention to finding something. Depending on the nature of your search, the DM might have you make a Wisdom (Perception) check or an Intelligence (Investigation) check.

2. All other situations

In all other situations, where no creature is actively searching for traps, you consult the creature's passive investigation score. Passive checks are described in the PHB:

A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.

A creature's passive score for a particular check is determined as:

10 + all modifiers that normally apply to the check

So if any of the creature's have a passive investigation greater than the spell save DC of the snare's caster, they succeed in noticing the snare.