I understand that Tortoise and Hare's meeting concludes the existence of a loop, but how does moving tortoise to the beginning of linked list while keeping the hare at the meeting place, followed by moving both one step at a time make them meet at the starting point of the cycle?
This is Floyd's algorithm for cycle detection. You are asking about the second phase of the algorithm -- once you've found a node that's part of a cycle, how does one find the start of the cycle?
In the first part of Floyd's algorithm, the hare moves two steps for every step of the tortoise. If the tortoise and hare ever meet, there is a cycle, and the meeting point is part of the cycle, but not necessarily the first node in the cycle.
When the tortoise and hare meet, we have found the smallest i (the number of steps taken by the tortoise) such that Xi = X2i. Let mu represent the number of steps to get from X0 to the start of the cycle, and let lambda represent the length of the cycle. Then i = mu + alambda, and 2i = mu + blambda, where a and b are integers denoting how many times the tortoise and hare went around the cycle. Subtracting the first equation from the second gives i = (b-a)*lambda, so i is an integer multiple of lambda. Therefore, Xi + mu = Xmu. Xi represents the meeting point of the tortoise and hare. If you move the tortoise back to the starting node X0, and let the tortoise and hare continue at the same speed, after mu additional steps the tortoise will have reached Xmu, and the hare will have reached Xi + mu = Xmu, so the second meeting point denotes the start of the cycle.