Is it that hot in Taiwan in the month of June? If its that hot, what do the locals do to avoid the heat wave? (I am a traveller/ Tourist)?

It's very warm in June time, and many days during the month can be hot especially between noon time to the late afternoon.

You can use MRT (subway like trains) to travel around Taipei city or go to Tamsui's fisherman wharf. The MRT stations and the trains are air conditioned.

There are many large malls around the city that have movie theaters, coffee shops, bookstores, restaurants, etc.

There are also Taipei 101 building, and the national palace museum.

There are many places sell shaved ice, cold drinks, ice cream, juice, and fresh fruits.

During the evening, when after it's cooled down a bit, you can choose to go to night markets, or bars.

Stay hydrated. Bring water with you.