Ponte is the informal command form of the reflexive verb ponerse.

ponerse = to put on oneself

me pongo — I put on myself
te pones — you put on yourself
se pone — he/she puts on herself
nos ponemos — we put on ourselves
se ponen — they/you all put on themselves/theirselves

Since we want to say “put on…” and not “you put on…” we need to use the command form of the verb, which is telling our children what to do (ponte) versus telling them what they are doing (te pones).

Another note: It would be INCORRECT to say “Ponte tus zapatos.” Please note that by using the reflexive ‘te’ you are implying “your.”

Ponte el abrigo.
Ponte los calcetines.
Ponte el pijama.
Ponte la gorra.