This is the 5384th most frequent Turkish word.
konuşmaya
"To the act of speaking" or "to speaking."
Here, 'konuşmaya' is the dative form of 'konuşma' (noun: speaking or speech), used to express continuing an action, ‘continued to speaking.’
Arkadaşlarıyla uzun bir süre konuşmaya devam etti.
He/She continued to speak with friends for a long time.
In this context, 'konuşmaya' is used with the verb 'başlamak' (to begin), indicating the initiation of the act of speaking.
Konuşmaya başladığımda herkes sustu.
When I started speaking, everyone went silent.
Here, 'konuşmaya' shows the act of trying to speak, as it is paired with the verb 'çalışmak' (to try).