The verb 'STOP' can be followed by 'the gerund (ING)’ or 'to + infinitive' but the meaning is different. Note:

"He stopped EATING lunch = he did not continue to eat his lunch. He finished eating. 

"He stopped TO EAT lunch" = he stopped what he was doing in order to then have lunch. 

Therefore: 

STOP + VERB ING = to no longer do something. They stopped talking to each other; they don't talk to each other anymore
STOP + infinitive = stop one thing to do another. They stopped to talk to each other; they stopped what they were doing to talk.