Why do phones often have HDR when some compact cameras do not?

There are two main things going on here. First, a smartphone camera has an advanced processor that can easily have the HDR software implemented for it. It isn't using a highly specialized chip and the cost of the device is often higher than the cost of entry level DSLRs despite the much simpler camera. This makes it a much more capable computing device and thus much easier to provide HDR functionality in device. To support this, it is actually quite common to find HDR settings on higher end DSLRs and point and shoots that have more powerful image processing chips.

The second issue is need. A smartphone generally does not shoot RAW images and has a much more limited dynamic range (the difference between the brightest and darkest part of the scene it can capture). Images also tend to be used right away for posting online or printing rather than being sent through a computer first. To get around these limitations, it is much more necessary to be able to process the image on the phone and expand the dynamic range. HDR is part of the solution to this problem.

Conversely, DSLRs almost always allow shooting RAW and even some point and shoots support it. RAW can effectively capture a wider dynamic range than JPEG usually can hold so it makes part of it unnecessary. Additionally, since images from stand alone cameras will often be processed on a computer, it is far simpler to provide simple exposure bracketing functionality that will allow the necessary HDR processing to be done later. Since nothing is really lost (other than some convenience) by postponing the processing to the computer, it is a cheap and easy thing to cut when trying to keep device costs low.