Let’s start with a picture. I’ve labeled various aspects using the words in the description you provided.
Source: CATSIP: California Active Transportation Safety Information Pages
The first sentence is written in terms of progression in the direction of traffic. The line is thought to “break into a dotted line [. . .] before it reaches the corner” because someone observing the line while traveling beside it in the direction of traffic would first see a solid line, then a dotted line, then (according to your excerpt, but not shown in the picture) no paint, all before reaching a corner. The word “corner” is sometimes used to refer to an intersection, abstracted slightly from the actual corner formed by the junction of the two roads.
The second sentence is a little tricky because “a bicycle lane follows specific width requirements” makes it sound like the bicycle lane is taking an action. In reality, it follows (i.e. conforms to) a specified width by design. This “specified width” is usually decided by a governing body of some kind. Where I live, the standards for these measurements are set by the Illinois Department of Transportation and published in documents such as this:
