To illustrate, let's use a simple example. Let's assume you need a logging feature in your application and your language/framework doesn't provide one(pretty much impossible nowadays).

The simplest solution is to just append to a hard-coded file. It solves the imminent problem. It's also very bad solution unless you will never write log again.

Instead, you build a Logger class/module/whatever. The logger provides 'write' functionality, which is abstracted away - client(i.e. code using logger) does not know or care how data is going to be logged. It's a responsibility of the logger to do that and make decisions on how to do that. Now you have abstraction - you depend on ability of Logger module/class/whatever to do its job, without knowing/caring/influencing actual implementation.

Now, let's talk about leaking abstractions. Let's assume you're need to differentiate between different log levels(info/warning/error). Logger is provided as a library(so not easily modifiable) and only has generic 'write' method. Proper solution would be to create a new abstraction, LoggerWithLevels, which would provide methods like 'write_warning', 'write_error' etc. Underneath, it would probably call just call Logger, prepending severity string to logged message. The important thing though is the fact, that we still base our code on abstraction - we don't know how our new logger represents different levels - we use appropriate methods and we assume it's handled alright.

What will likely happen though is that programmer will not create new abstraction layer. Instead, he will start manually or semi manually format strings sent to a logger. Effect will be the same, but abstraction will be now 'leaking', because severity level is now dependent on implementation(prepared log messages), not on logger API. Therefore, any change in the process(like saving logs to online service instead of disk file) will now require changing code in multiple places instead of just one.