I have a player who is playing the above race/class and noticed that via the Variant Tiefling Feature they are given:

Devil’s Tongue. You know the vicious mockery cantrip. When you reach 3rd level, you can cast the charm person spell as a 2nd-level spell once with this trait. When you reach 5th level, you can cast the enthrallspell once with this trait. You must finish a long rest to cast these spells once again with this trait. Charisma is your spellcasting ability for them. This trait replaces the Infernal Legacy trait.

Hellfire. Once you reach 3rd level, you can cast the burning hands spell once per day as a 2nd-level spell. This trait replaces the hellish rebuke spell of the Infernal Legacy trait.

Firstly, there is a space missing between 'enthrall' and 'spell', but my question is how can I add these spells to the character's spell list? He is 3rd level but they don't appear to be available in the spell list and I'd like him to see that he has access to these.

You have to make sure you choose the variant Tiefling race rather than the normal Tiefling and then you can choose one of those features in the Race section of the builder

Yes, that's what he's done, but those spells can't be added (or tracked) via the spell list.

And did you notice the bit that says, "Devil’s Tongue, Hellfire, and Winged are mutually exclusive"? 

No I didn't, thanks for pointing that out. I thought it seemed a bit OP.