Can someone help me understand how a series of interest rate call options with different maturities and the same exercise price can be combined to form an interest rate cap?

And how an interest rate floor is a portfolio of interest rate put options?

You need to look at the definition of a cap (and a floor).
Caps and floors are long-term derivatives, and their lives are broken up into a series of shorter periods.
With an interest rate cap, the buyer receives payments at the end of each period in which the interest rate exceeds the agreed strike price. The strike price is the same for each period/
You can replicate the payment stream of a cap if you write a separate interest rate call option (a caplet) for each period, then the payoff of the call will be the sum of the payoffs of all the caplets.

Similarly, a floor can be replicated by writing a separate interest rate put option (a floorlet) for each period