Both would be translated to the same sentence in English
I was looking for my dog yesterday
but, in order to explain what estar
is doing to the sentence, ... you have to remove the status of looking for
Yo buscaba a mi perro ayer.
I looked for my dog yesterday
there is still a continuity
VS
Yo estaba buscando a mi perro ayer
I was looking for my dog yesterday
Both estaba buscando
and buscaba
are in the imperfect tense, meaning there is no termination of the event.
Estar means to be in a current, specific, and particular state of time or place. So to say
Estaba buscando a ....
means, I was in a state of searching
.. not I searched
..
Yo estaba buscando a mi perro todo el día de ayer.
I was in a state of searching for my dog all day yesterday
VS
Yo buscaba a mi perro todo el día de ayer.
I searched for my dog all day yesterday.