How isomorphism makes groups alike? If two groups are isomorphic to each other they are same ( with different elements). But my question is when someone came up with the concept of isomorphism he or she must had checked some basic facts about those groups. But there are so many properties of groups; so on the basis of those few facts how could he tell that those groups were same?

Anything that is a group property must be determined by the group's structure (which consists of a set and an operation).
So if two groups have indistinguishable sets and operations, they must be indistinguishable as groups (i.e. isomorphic as groups).
An isomorphism is a bijection (i.e. one-to-one and onto) function, so the underlying sets of two isomorphic groups are essentially the "same". 
For example: $f:\{1,2,3\} \to \{a,b,c\}$ where $f(1)=a$, $f(2)=b$, and $f(3)=c$. Even though the sets $\{1,2,3\}$ and $\{a,b,c\}$ are clearly different (one has numbers and the other letters) in a different way they are essentially the same as sets (they both are collections of 3 things). The "difference" between the sets is about being "letter" or "number" which aren't really "set" properties. So in some sense set theory is largely blind to the "differences" between my sets.
An isomorphism also preserves the operation: $\varphi(x\star y)=\varphi(x)*\varphi(y)$ if $\varphi$ is an isomorphism between two groups where the first group has an operation denoted $\star$ and the other group's operation is denoted by $*$.
This operation preservation guarantees that the two groups' Cayley tables match (if we had finite groups where we could write such a thing down). In words it says that if we operate then map, we will get the same answer as if we map then operate. 
This is enough to make sure that anything "group theoretic" said about one group will necessarily be true about the other group. This is because "group theoretic" statements just refer to the set (which we have made sure look the same) and the operation (which we have made sure operate the same).
I hope this helps!
