It's actually two idioms tied together. "To cut corners" means to leave the some of the details of a task unfinished while doing the main portion of the task. If you cut corners while cleaning your house, you may sweep the main part of the floor, but not underneath the table. If you cut corners on a paper for school, you finish writing the paper, but you don't edit it or you only write 4 pages when the assignment was for a 5 page paper. Doing something "left and right" means to do a lot of that thing. A common phrase is "Spending money left and right" which means you are spending a lot of money. So, "cutting corners left and right" means that the person is doing as little as they possibly can (and probably a little less than that) to call the thing they're working on complete.

The first part, to "cut corners," means that someone is taking a shortcut or doing something in the easiest, cheapest, or quickest way possible. Oftentimes, when someone cuts corners they also end up doing low quality work so the phrase usually has a negative connotation. The second part, "left and right," means that it is happening everywhere or very frequently. So when someone cuts corners left and right, it means that he or she frequently does things the easiest, cheapest, or quickest way possible instead of doing things in a way that requires patience or hard work. You can also use each phrase independently. For example: "You're always trying to find ways to cut corners in life. If you really want to learn a new language, you need to stop buying those programs that guarantee fluency in only two weeks and start practicing speaking regularly." "We dominated at soccer yesterday. We were scoring goals left and right."

Thumbs up to Angela. In the UK, we say "right, left, and centre" in this context, instead of "left and right". It's useful to know the US equivalent.