Please teach me how to use ‘hot diggity’, and the meaning of ‘corkscrew twists’

"Hot diggity" is an expression from the 1950s. People use it today slightly more often than they use "23 skiddoo" (an expression from the 1920s). The reviewer is using it ironically or archly; if he were speaking to you he would probably arch his eyebrows and/or theatrically change his tone of voice to indicate an anachronism.

"Corkscrew twists" refers to plot twists. A corkscrew is a tool for pulling the cork from a bottle, and has a series of twists in it. When used figuratively to refer to a plot, it means the twists leave you spinning.

Suitable replacements for "hot diggity" would be:

• gee whiz!
• gee willikers!
• oh boy!
• oh my!
• hallelujah!

The meaning is more of a retro way to express excitement or gleeful expectation. "Oh my goodness!" is similar but tends to be more of an expression of surprise than excitement.

The phrase and its ilk are not often used unless expressing excitement towards something from the era when the phrase was more common. In this case, the '60s.

"Corkscrew twists" is interesting because of "corkscrew" which refers to anything extremely curvy or twisted. A loose definition could be "something that consistently and constantly turns, twists or weaves." It is used to describe a great many, many things and can mean something that figuratively or literally twists:

This road is a corkscrew.

The dog's path corkscrewed across the lawn.

My face corkscrewed in agony.

A story's twists are its plot twists and a play that corkscrews would be one that twists a lot. The specific meaning of the quoted phrase is that the inspiring source material has a great many plot twists but the play itself is not confusing or hard to follow — the audience will not be left wondering what just happened.

How popular hot diggity is in day-to-day usage?

It's not at all popular in day-to-day usage. It's an antiquated saying, as Robusto and MrHen have noted. It's something you might expect to hear in an old black and white movie or from an excited elderly person, but it would rarely, if ever, come up in casual conversation of young or middle-aged persons.

A variant to hot diggity is hot diggity dog.

Etymology Online doesn't have any entry on hot diggity but I did find this entry on Wikipedia on the song Hot Diggity (Dog Ziggity Boom):

"Hot Diggity (Dog Ziggity Boom)" is an American popular song written by Al Hoffman and Dick Manning. Published in 1956, it was recorded by Perry Como and went to #1 on the Billboard pop music chart later that year. ... The phrase "hot diggity dog!" dates to at least 1928, when Al Jolson was recorded saying "Hot diggity dog! Hot kitty! Hot pussycat! Didn't I tell you you'd love it?" after a performance of the tune "There's A Rainbow Around My Shoulder".

Hot diggity dog is also a not uncommon name of hot dog restaurants around the United States.