From the Downtime Activities on page 187 of the PHB:
You must be proficient with tools related to the object you are trying to create (typically artisan’s tools). [...] For example, someone proficient with smith’s tools needs a forge in order to craft a sword or suit of armor.
So RAW it would appear that crafting doesn't actually require any ability checks. You just need proficiency in artisan's tools to be able to craft non-magical items in the first place.
But the DM might call for other ability checks that require artisan's tools.
From page 154 of the PHB:
A tool helps you to do something you couldn't otherwise do, such as craft or repair an item [...] For example, the DM might ask you to make a Dexterity check to carve a fine detail with your woodcarver’s tools, or a Strength check to make something out of particularly hard wood.
Plan A is to go into the dungeon and swat at the monsters with our swords until they fall over. A lot of adventurers do well with Plan A for a long time.
Plan B kicks in after the swords are ruined by a Rust Monster, the secret door is jammed shut because the hinges are seized, or you realize that the only way past the guards requires a wheelbarrow and a holocaust cloak and there is a broken-down cart under that big tarp behind the shed.
If only "Carpenter's Tools" had been listed among your proficiencies you could have totally made that last plan work.
In order to use the artisan's tools, he needs proficiency in them. If he wants to craft any sort of object using the tools he first needs proficiency.
You might also apply the proficiency to checks made for non-crafting purposes. For example, if your player's character is a smith, in addition to crafting or repairing an item, he may wish to inspect the quality of some weapons or tools or identify their origin. You could apply the proficiency in the tools to the ability checks made for these purposes.
I've seen a very specific example where a character was rewarded a runestone that someone attempted to, using smith's tools, affix to a weapon that would imbue the weapon with a specific enchantment. To do so, the DM gave the Forge Domain Cleric a high DC to meet that was a STR/DEX roll + their proficiency bonus because they were proficient in smith's tools, being a career blacksmith. The affix attempt could be made once per day (or twice/day during downtime) until successful. Only if proficient with those tools, one could also attempt to remove the runestone in the same manner with the same check at a higher DC, though a low roll could destroy the item in the process.