In my first "basics" post on connections, I explained why a network that requires connections (or as North Americans call them, "transfers") can actually get people where they're going faster than a network that tries to avoid them.
But there's another important reason to plan for connections rather than direct service, one that should be important to anyone who wants transit to be broadly relevant to urban life: Unless you welcome and encourage connections, your network will become very, very complex.
If a transit system is too complex for people to learn, it can't be used spontaneously, and if you can't use transit spontaneously, then transit is reduced to a very limited role in urban life, serving only trips that people make regularly and can therefore learn how to make even in a complex system.
In presentations on this point, I often compare a network based on connections -- such as that of Los Angeles -- with a network based on fear of them -- such as the inner network in Sydney. Here are more or less comparable slices of the two networks, each image about 8 km wide and 3 km high. Both lie just south of the CBD of each city, so you can see routes converging toward the CBD on the north edge of the map.
Network maps drawn by transit agencies are often hard to compare, because each is full of details and conventions that make sense only locally. To make sense of the Los Angeles map, you must know that line numbers in the 300s and 700s are limited-stop or "Rapid" services that almost always appear on top of a local-stop service on the same street. So for example on Vernon Avenue, the big east-west corridor across the center of the Los Angeles image, 105 and 705 are local-stop and rapid versions of the same line. These two services are not duplicating each other; in fact, they are designed to complement each other. So you should think of 105 and 705 as a single composite line.
Once you account for that, the striking thing about the Los Angeles map is that most streets have only a single line on them, even though we are immediately south of downtown and you would expect many lines to be converging.
By contrast, in a similar sized slice of Sydney, most streets have several lines on them, and in most cases these lines are not designed together. In Los Angeles, if you see two buses right next to each other for many miles along the street, they are probably on the same line and one of them is probably late (a phenomenon called bunching.) In Sydney, if you see the same thing, the two buses may well be scheduled to do that, usually because they're on different lines that run on top of each other on the same street, and no attempt has been made to co-ordinate their schedules. For example, Sydney's Oxford Street, for the entire 3.4 km between Taylor Square and Bondi Junction, has four local bus lines and one limited-stop line, ranging in frequency from 10 minutes to 30 minutes, with no particular co-ordination of their schedules. On this street, bunches of buses are a common sight because in many cases, they're scheduled that way.
Sydney-style networks happen where the prevailing assumption is that people will resist connections at all costs, which means that there is a strong motivation to run direct services between every possible origin and destination. Even where planners would like to introduce connection-based design, they are frustrated by a fare structure, out of their control, which penalizes connections. So Sydney ends up with lots of overlapping lines, a high degree of complexity.
This problem is also the cause of others, such as the excessive volume of buses, most of them not full, choking the narrow streets of the CBD. Obviously, if you didn't feel constrained to run direct service from everywhere to the CBD, you'd organize some buses to feed others (or to feed trains). This would give you fewer buses in the CBD, with heavier loads on those buses, and thus a stronger moral claim to transit priorities, such as exclusive lanes.
Just for fun, here's an 8x3km slice of an even simpler network, also just south of downtown, though admittedly in a smaller city. Portland:
I often show this map in Australia because the pattern is even more clear than in Los Angeles. The simplicity is extreme. Every arterial has exactly one bus line on it, with a simple 1-2 digit number. The only exceptions are very close to downtown, where radial lines inevitably have to converge, and a few ghostly line numbers that appear on a white background, which signifies a handful of trips that run only on the peak. (Note, by the way how this notation helps the viewer see the all-day network on which she might rely,. This clarity is quite unavailable on the Sydney map. The Los Angeles map also doesn't distinguish peak-only services, though across much of the city there are very few of them.)
The Portland map also makes clear that even in a city with an exceptionally strong downtown, many frequent lines don't go there, including 70, 71, 72, and 75 on the map above. The focus, as in Los Angeles, is on anywhere-to-anywhere travel using connections. Because these "crosstown" lines rely on connections more than the radial lines do, they all run as frequently as possible, and as late into the night as possible. Sydney, too, has non-downtown lines, but with one exception, the 400, they are too infrequent for fast connections to be possible, and often shut down earlier than the rest of the network. Again, the frequency is hard to afford in Sydney because resources are divided over so many lines, even those these lines are often overlapping. Portland and Los Angeles, by contrast, are running the minimum possible number and length of lines, so that they can run each one as frequently, and as late, as possible.
Sydney planners will sometimes argue that Los Angeles and Portland are simpler because they already have street grids, while Sydney is an unplanned accretion with no large-scale order to its street network. This is partly true; grids are certainly easier to remember, in both cities and transit systems. But you can take a structure like Sydney's and design a more gridlike system, featuring fewer and therefore more frequent lines, all designed to work together to serve any origin-destination pair, rather than picking a few major destinations as today's network tends to do. I did a sketch of one here.
UPDATE: An interesting series of comments, starting here, is applying this thinking to equally gridless London, including a very, very frequent grid bus network sketch by commenter Zoltan, which is here. Remember that polar or "spiderweb" grid, consisting of radials and circles, can also achieve good connectivity while still focusing on a center. Most workable grid networks, including my sketch for Sydney here, are a hybrid of polar grids and standard rectangular grids.