Learn Spanish (Latin America): Yo me iba a meter el cono en la boca, pero me manché la camisa. - I was going to put the cone in my mouth, but I stained my shirt.

The structure ir a plus an infinitive uses the imperfect (iba a meter) to express an intention you had in the past ("I was going to put"), which is usually unfulfilled or interrupted by another event. If you used the preterite (fui a meter), it would mean you actually went somewhere and completed the action ("I went to put").

Why does it say la boca and la camisa instead of mi boca and mi camisa?

In Spanish, when talking about body parts or items of clothing you are wearing, you generally use the definite article (el or la) rather than possessive pronouns (mi, tu, etc.). To show whose mouth or shirt it is, Spanish relies on the pronoun placed earlier in the phrase—in this case, me (me iba a meter, me manché).

Could the first me go at the end of meter instead?

Yes, you could perfectly say Yo iba a meterme el cono en la boca. When you have a conjugated verb followed by an infinitive (like iba a meter), pronouns can either go before the conjugated verb (me iba a meter) or be attached directly to the end of the infinitive (iba a meterme).

Why use meter here instead of for "to put"?