In the definition of animal species we say they can't have
sexual intercourse but we see the tigers and lions can have sex and have cubs.
How can this paradox be explained?

The accepted species definition is the ability to produce viable offspring rather than the ability to mate.

see
http://evolution.berkeley.edu/evolibrary/article/evo_41
https://en.wikipedia.org/wiki/Liger

That is only one definition of about 15 that are in common use. Biologists use different definitions for different sitautions - being capable or reproducing is not much to go on when dealing with fossils or asexual organisms and plants for example. The reason we have so many is that life is very variable and the concept of a species is always going to be somewhat artifical - individuals and lineages are not discreet entities like atoms, but are part of an ever evolving chain. None of the individuals that make up lions as a species were alive 100 yeas ago and none of the ones alive now will be around in a hundred years but they are all part of the same species. Speciation can take time and species can merge into one another or are not yet properly split so we need apprpraite definitions to deal with this pheonmeon.