The why of frugal homeschooling is the easier of the two to answer. The why is that you (if you're frugal-homeschooling) have limited funds, your family is probably living on one income, or at least less than two full-time incomes, so that somebody can be home to homeschool.
Or maybe you just like a challenge.
The short answer of "how" is "don't spend much money." But since that's also a silly answer, I'll try to expand that into something more useful.
1. Use what you have.
2. Use what you have creatively.
3. This is the hardest part to explain: stay aware of your "big picture." Unless you're naturally serene about letting the unschooling chips fall where they may, you need to keep evaluating, planning, trying to keep in mind whatever educational goals or philosophy you steer by. Plus whatever family circumstances, special needs, etc. you have to deal with.
4. In other words, you can use what you have, or what comes your way, as long as it fits into your overall education plan.
In Lloyd Alexander's book Taran Wanderer, the main character Taran meets Llonio, a father who supports his family by taking hold of anything that fate throws in his net--literally. The family never knows from one day to the next what will float down the river, but they cheerfully take whatever comes, and eat it or wear it or use it. As Taran stays with Llonio's family, he appreciates their generosity and their creativity, but he also eventually realizes that their way of life is not exactly for him. He wants to do a little more purposeful seeking, instead of just catching what comes his way.
I think there's room for both, even in a frugal lifestyle and in frugal homeschooling. When I wanted to make a particular doll from a particular pattern, I kept my eyes open for certain colours and fabrics. I never did get to the outlet store that sells rug yarn, but I found something pretty close that also worked. When I crocheted monkeys last Christmas, I bought yarn in the right colours. On the other hand, I've sometimes started with a piece of fabric or a ball of yarn, and asked "what could this be? How big is it, how much of it is there, is there enough for this or that? What else would it work with? And what do we need right now, who still needs a Christmas gift?"
The same principles apply to menu planning. What's available? What's the weather like? What sort of meals does your family eat? What do you need to add to the shopping list to turn wieners and cauliflower into a meal? What's still a favourite, what's getting old, and what new things have you been wanting to try? Sometimes you go shopping intending to buy chicken thighs, because somebody gave you a new recipe, and that is what you bring home. Or you look in the freezer, and that's what's there. Or it could happen that chicken is too expensive, so you buy something else.
One useful exercise to strengthen frugal homeschool muscles is to pretend you are (or maybe you really are) in a situation where, for whatever reason, you are suddenly limited to a few books and resources. It could be a Bible, dictionary, telephone book kind of thing; or you can go with a more random choice, like the stack of books you just brought home from the library. From very loose planning ("read the book"), to more structured copywork and dictation, notebooking, dramatizations, or complete unit studies, how many ways can you think of to get the most out of this resource? If it's a map, are there ways you could add tags or markings to illustrate something you're studying? If it's a math activity book, which activities can you honestly imagine doing, and (just as important), which ones will provide the strongest learning experiences for your children?
If it's a book of poems, how will you get the most of out of it? Have any of the poems been set to music? Have any actors recorded them? (Check out anything you can find by the First Poetry Quartet.) Are there possibilities for acting them out? (Never underestimate the potential for this--I still remember the Apprentice dramatizing Blake's "A Poison Tree," including the enemy's death throes.) Can you use any of Ruth Beechick's suggestions, such as turning verse into prose? Or can you use a poem as a jumping-off point for something original? Or you can just read a poem slowly and carefully, maybe taking turns on stanzas, copying or memorizing favourite lines. It's also educational, or just entertaining, to group certain poems together, maybe in combination with art, music, or other readings. Our church music director once did this as part of a holiday program: several people of different ages read winter-themed poems by Robert Frost. Can your students plan a "poetry concert," just for your family or for others as well? You can see where I'm running away with this...but that's the point, that you can take any worthwhile book as far as you like, use it as far as you can, and it won't cost you any extra.
Linked from the Festival of Frugality #357, and from the Carnival of Homeschooling: End of the Road Edition.