。
She started putting on weight.
Literal
She [topic-は] becoming-fat began.
太りはじめた is a compound verb 太る ('to put on weight, gain weight') + 始める ('begin'). Note that 太る itself is a change-of-state verb, so the bare 太った past form means 'has gained weight / has become fat,' while ~始める marks the onset of the gaining process. 太る's stative ~ている partner 太っている means 'is fat / is overweight.'