The expression "gone to seed" usually has a negative connotation, meaning disheveled, declining or otherwise past-prime. When your 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 anthropomorphism aside, the plant's formerly edible parts become shriveled, bitter and woody. This can look like a bad end for your endive patch, but crops that have gone to seed can still play an important role.
A plant that has run its productive course 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 nearby plant. (For this reason it's inadvisable to replant seeds from garden squash or melons, because squash-family hybrids can be foul-tasting or poisonous.) The seeds might also be sterile and not sprout at all.
The practice of seed saving is another discussion, full of complexity and art, and it's more commitment than I care to take on. Instead, when the greens bolt, 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 tomatoes is fine with me. I'd much rather have edible crops volunteering themselves than many of the weeds I know.
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. These elders crowd the garden with their blossomed stalks, providing cool, moist shade that allows later-blooming plants to maintain their tender youth a little longer. Newly sprouted plants are thus sheltered as well.
A plant that has run its productive course is, naturally, a source of seed.
Perhaps most importantly, the flowers attract pollinators. The nation's bee population is plagued by mites, viruses and neonicotinoid-based pesticides, all of which have been implicated in die-offs, and 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 a gridlock of bee traffic, which helps transform it into place of interaction between interdependent segments of the greater ecosystem. Watching bees patronize the newly opened blossoms is never a buzz-kill.
A garden gone to seed is a diverse ecosystem, with young plants growing in the shade of old-growth annuals, 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 home gardens, however, are about production alone. It’s probably more pragmatic to spend your time at a paying job instead of gardening and buy your food. But a little plot of land feeds 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 stroll among your plants is like entering the forest to see what you can find, and when you push the bolted radicchio aside to discover a young head of hidden lettuce, it feels like a gift.
I take care throughout the season to create this kind of luck by casting handfuls of seeds every which way in spring and summer. Mostly I toss seeds for greens, herbs and carrots, all of which can be planted all summer long 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’d scattered their seeds there, or allowed their parents to bolt, doesn't diminish the thrill. But as you'd expect, hurling seeds randomly in 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 others. Bolted plants might block a sprinkler or attract unwanted pests and birds that shit on your parsley as they eat cilantro seeds. You may choose to let only a select few bolted plants stick around.
Managed chaos probably doesn't fit most people's idea of what a well-cared-for garden should look like. And to be honest, while it looks good to me, I often find myself slightly embarrassed when I show my garden to others. I feel compelled to blurt out why I chose to let it go to seed. In the end, gardens, like pets, often come to reflect their owners. Some gardeners value appearance above all else. Others care more for an interesting personality. Then there are those like me who use the space to explore the greater ecosystem, and as a source of adventure and surprise. The bolted variety of gardening may not pay for itself in the cold economic calculations of input versus return. But as entertainment it's cheaper, and more enlightening, than a trip to the movies.