Dó is more "to him" than just "him" - it's the 3rd person (masc.) equivalent of dom or duit. The oddness in this translation is the ligeann do construction. It would be ligim do Sheán teacht, for example, if you wanted to say I let Seán come.
(It might be helpful to remember that in English, this "let" has an implication of "give permission to", so the use of dom, duit, dó etc with lig isn't necessarily that strange).
lig do is covered in the FGB.
Note that English is ambiguous, "I let myself into the house" can be past tense ("I let myself into the house yesterday") or present habitual ("I let myself into the house every evening"), so you would need lig mé for the past tense, and ligim for the present.
You don't need a preposition with lig isteach, so you will encounter both ligim mé féin isteach and ligim isteach mé féin, for "I let myself in", but for "into the house", you would use isteach sa teach (not "isteach an teach"), and the mé féin would generally go first, I think.
Lig mé mé féin isteach sa teach inné
Ligim mé féin isteach sa teach inné gach tráthnóna