The height of a zebra is lesser than giraffe.

Correct sentence: The height of a zebra is lesser than that of a giraffe.

In this sentence, two different heights are being compared - that of a zebra and that of a giraffe. Only "that of a giraffe" properly indicates that two different heights are being compared.

Like yesterday, the sports page of today’s newspaper was filled with details of the World Cup match.

Correct sentence: Like yesterday’s cover, the sports page of today’s newspaper was filled with details of the World Cup match.

What is being compared are not the two days ("yesterday" and "today"), but rather the content displayed on the sports page of each day’s newspaper. Without clarifying this fact with "yesterday’s cover," the sentence inappropriately compares the day "yesterday" with the sports page of today’s newspaper.

After working for so many years, the chef declared that Jenny’s ability to make delicious soups was better than anyone else.

Correct sentence: After working for so many years, the chef declared that Jenny’s ability to make delicious soups was better than anyone else’s skill.

As written, the sentence actually compares this ability with other people (as opposed to those other people’s abilities to make delicious soups). Only by comparing "ability" with "skill" does the sentence make logical sense.