The expression "gone to seed" usually has a negative connotation, meaning disheveled, declining or otherwise post-prime. When garden plants go to seed, or "bolt," they quit being what you planted and become gangly towers looming over the garden. In this respect they're more like teenagers than elders, but anthropomorphisms aside, the plant's formerly edible parts are now shriveled, bitter and woody. While this can understandably look like a bad end for your endive patch, crops that are going or have gone to seed can still play an important role in the garden.
A garden plant that has run its course and produced seeds is, naturally, a source of seed. Depending on the plant's propensity to crossbreed, the seeds it produces might be true to the parent or a mix of parent and some similar plant. Or the seed might be sterile and not sprout at all.
Instead of saving seeds, I simply let the seeds fall where they may. If any of them happen to sprout next month or next spring, great—any time a yummy plant wants to grow up between my garlic or tomato plants is fine with me.
The first plants to bolt are generally the leafy cold-weather plants like spinach, lettuce, escarole and cilantro, and they make their moves when spring turns to summer. The old farts crowd the garden with their blossomed stalks, providing cooler, moist shade that allows the later blooming plants to maintain their tender youths a little longer. Newly sprouted plants are thus sheltered as well.
Perhaps most importantly, the flowers attract pollinators. The nation's bee population is currently plagued by mites, viruses and neonicotinoid-based pesticides, all of which may play roles in colony collapse disorder. This dire situation is alleviated, if only slightly, by a pesticide-free garden gone to seed, which can provide a safe, nourishing haven for imperiled bees. My bolted garden is practically a gridlock of bee traffic, which helps the garden become a place for interdependent parts of the ecosystem to engage. Watching bees patronize the newly opened blossoms is hardly a buzzkill.
A garden gone to seed, with young plants growing in the shade of old-growth annuals, is a diverse ecosystem, but diversity has its drawbacks. Bees aren't the only critters that prefer dense polyculture to boring monoculture. Disease and pest problems may increase. As your plants fill out and crowd together, they won't be able to grow as large as they would if they had more room. If your singular goal is to pull as many pounds of food from the garden as possible, then letting plants bolt might not be the game for you.
Few gardens, however, are about production alone. With food prices as low as they are, it would probably be more pragmatic to spend your time at a paying job, instead of gardening, and to buy your food. But gardens feed more than just your belly. A diverse garden, with different types and ages of plants, is interesting. A garden gone feral can blur the line between growing food and gathering it. A trip to the garden starts to feel like entering the forest with your basket to see what you can find.
I take care throughout the season to create this kind of luck by casting handfuls of seeds every which way. Mostly, I toss seeds for greens, herbs and carrots, all of which can be planted throughout the summer for a fall crop and all of which do well in partial shade. When I happen upon an unexpected parsley patch or grove of carrots, the fact that I had scattered their seeds there, or allowed their parents to bolt, doesn't diminish the thrill of discovery. Hurling seeds randomly at the garden won't result in orderly rows. Those who like their garden linear and neat should probably avoid this tactic.
There's a fine line between letting your garden go to seed and simply abandoning it. I call it "managed chaos." You may need to pull some plants that are crowding or shading certain other plants. Bolted plants might block a sprinkler, or attract unwanted pests or birds that crap on your parsley as they eat cilantro seeds. You may choose to let only a select few bolted plants stick around, like islands in an otherwise orderly garden.
Managed chaos probably doesn't fit most people's idea of what a well cared for garden should look like. Some gardeners value appearance above all else. Others care more for an interesting personality. Some, like me, use the garden to explore the greater ecosystem, as a source of adventure and surprise. Gardening, especially bolted-garden gardening, may not pay for itself in the cold economic calculus of input versus return, but as entertainment it's a lot cheaper, and more enlightening, than a trip to Disneyland.