Learn Spanish (Spain): Me seco con la toalla. - I dry myself with the towel.

The verb secar simply means "to dry" (like drying a plate). Because you are performing the action on yourself, you must use the reflexive form secarse. The pronoun me tells us that the person doing the action and the person receiving the action are the same.

Would it be correct to say seco a mí mismo to mean "I dry myself"?

Not exactly. For daily routines, Spanish relies on reflexive pronouns like me rather than a literal translation of "myself". You can add a mí mismo for strong emphasis, but you still must include the me before the verb: me seco a mí mismo. In normal conversation, just me seco is perfectly complete.

Why does the sentence use la toalla (the towel) instead of mi toalla (my towel)?

You can say mi toalla, but Spanish frequently prefers the definite article (la) over possessive pronouns for personal hygiene items, clothing, and body parts. Because you already used me to show the action is happening to you, it is naturally assumed that you are using your own towel.