When using the word 'was', you are referring to something in a past tense, or something that happened before. Combining it with the word 'busy' you can express something that was occupying you in a past time.
Here are some examples:
I was busy thinking.
I was busy working.
I was busy cooking dinner.
I was busy talking on the phone.
I was busy cleaning the house.
I was busy studying for my test.
I was busy thinking of ideas for our website.
I was busy entertaining our neighbors.
I was busy completing my housework.
I was busy learning new things.
By changing 'was' to 'am' you change your message from past tense to present tense and refer to something you are doing 'now.'