I do not understand why the following holds: the larger the divded paid the lower the vaue of the call and the higher the value of the put

As dividends become larger, the ex-dividend price of the stock becomes lower. A lower stock price lowers the value of a call option and increases the value of a put option (i.e. call delta is positive, put delta is negative). You can also think of this mathematically in terms of put-call parity below:

C = (S-D) + P - X/(1+r)

or

P = C - (S-D) + X/(1+r)

As D increases, C decreases and P increases.

Thanks for the clear explaination