to observe attentively, to pay close attention.
Most often heard in the fuller idiom echarse la manta a la cabeza: to throw caution to the wind and take a risk without worrying about the consequences. Informal, mainly used in Spain.
Using the phrase without a la cabeza, which sounds incomplete. It is reflexive: me echo, te echas, se echa.
Want to practice? Try a conjugation exercise.
"echarse la manta" means "to observe attentively, to pay close attention.".