"Me puse a" is a weird phrase
Why does that mean "I started to?
2 Answers
It's just how it is lol - why do we say 'I got down to it' to mean I started to.
I think that ponerse a is often fairly close in meaning to 'get down to something' (not sure if you use that expression where you are) but rather than just mean 'to start' it includes some kind of feeling of determination to get stuck in and get something done.
To start without that extra nuance can also just be 'empezar a'.
'Ponerse a' also sometimes is used to describe something that happens suddenly or as an unexpected reaction to something else.
La niña se puso a gritar al ver la araña en su almohada.
The girl started to scream when she saw the spider on her pillow.
Me pongo roja.
Me pongo a llorar como una niña.