To satisfactorily answer why ditches are dug across a trail, we need to first consider another question: why do trails sometimes become ditches?
Trails become ditches because of erosion. Use of a trail, whether by boots, hooves, or tires, tends to erode a slight dip in the tread, sort of like a very shallow rut. Where trails "just happen" (like the original Cedar Butte trail) on a slope without construction of a properly drained tread, water flowing down-slope is caught in this rut, and starts flowing down the trail. Normally water flowing down a slope is slowed by vegetative debris and root systems, giving it a chance to soak in. On a trail there are no such natural obstructions, so the water gains in speed and momentum, greatly increasing its erosive effects. During a heavy rainstorm there may be a regular creek going down the trail, grinding away at the steeper sections.
When erosion is not controlled trails can become so deeply gullied that they have to be abandoned. A prime example is "Anklebreaker Gully" at the top of the first section of the Cedar Butte trail. On the map it appears to be an attractive short-cut. Especially when you are coming back a little late, and it is nearly dark. Don't take it! It is quite hazardous, especially going down hill. But if you are curious about what erosion can do to a trail (and this isn't even the worst possible case!), you might give it a careful exploration.
So these little ditches--more properly, "drain dips"--are cut across the trail to divert water off of the trail. On a well-designed trail these dips will be stretched out some ten feet or more, and so they seem quite shallow. On a steeper trail, or one that has tight constraints on where water can be drained, the dip will be shorter, which for the same depth will make it seem more like a small ditch.
There is a bit of an art to these things. Not only do they have to be graded properly so they drain, and don't make a puddle in the trail, the also have to be big enough to not fill up with silt. And they are usually angled to gently persuade the water to turn aside rather than run up and over the dip, and on down the trail. This requires some idea of how much water will becoming down the trail, and how fast, and how much silt is being carried down. The type of soil also makes a difference, and even whether the spot is sunny or shady. Getting it wrong can easily make things worse.
"Water bars" or "check dams" of either wood or rock can also be used to divert running water off of a trail, but these have generally fallen out of favor as they are prone to being torn out. There is evidence that some bikers were tearing them out in the mistaken belief they were intended as bicycle obstacles.
You may have noticed the preponderance of drain dips--and steps--in the very first section of the Cedar Butte trail. Most of the rest of trail has been re-routed and re-built with particular care to be more durable. But in the first section the difficulty of building new trail in heavy brush and logging slash, coupled with the difficulty of weaning the more persistent users off of the areas needing restoration, made it preferable to try to harden the existing trail. However, this section still requires much maintenance (such as regularly clearing out the drain dips), without which it would eventually become just as eroded and hazardous as Anklebreaker Gully.
Drain dips are just one way of dealing with certain kinds of problems. Even better is to build trail to avoid such problems in the first place. A large part of this is in properly the routing the trail, with careful consideration of grade, soils, and other factors. There are many subtle factors in making good trail that will last without a lot of maintenance. But trails that "just happen" are usually deficient in many critical respects. If your favorite short-cut (or any other trail) has been blocked off, it is most likely because of potentially serious problems. Please respect those closures! Sure, your single passage might not do much damage. But the problem is that where one person goes, many others will follow. Building and maintaining trails takes a LOT of work, and almost entirely by unpaid volunteers. If you like good trails, and want to keep them: please stay on them!
(If you want to go cross-country--which can be a lot of fun--please stay away from any trails, and avoid any stretches that are starting to show signs of becoming trail. And especially--please!--avoid any old trails that are being revegatated. This is necessary to prevent the formation of bad trails that eventually become big bad gullies.)
Short-cutting switchbacks is especially destructive, as it damages the trail at a vulnerable location that is usually very difficult to repair. So please do not take shortcuts. And don't let your kids take shortcuts! (Explain to them why.) And if you see anyone else taking shortcuts: yell at them! (Tell them you are doing them a favor, lest they be set upon by irate trail trolls with iron-tipped tools!)
Having come this far, perhaps we can throw in one final request: please be careful with the edge of the trail! The weakest part of a trail going across a slope is its outside edge. It takes only a few steps on that edge to break it down, and then the edge starts moving inward, where it finds a whole lot more traffic. And pretty soon everyone is slipping down the side and paring away tread. So try to avoid walking on the vulnerable outside edge of the trail. If you have to step off the trail--perhaps to let a bear go by?--it is generally preferable to step up hill. If you come across some trail builders hard at work, it is better--and safer--to pause for a moment and wait for them to stand back then to go charging around them.
If you would like to learn more about trail building, or perhaps even help, check with your local trails clubs or other hiking groups. Many of them have regularly scheduled work parties, and your help is welcome even if you have no prior experience.