The Elements of Style · Rule 6

Do not break sentences in two.

In other words, do not use periods for commas.

I met them on a Cunard liner several years ago. Coming home from Liverpool to New York.
He was an interesting talker. A man who had traveled all over the world and lived in half a dozen countries.

In both these examples, the first period should be replaced by a comma, and the following word begun with a small letter.

It is permissible to make an emphatic word or expression serve the purpose of a sentence and to punctuate it accordingly:

Again and again he called out. No reply.

The writer must, however, be certain that the emphasis is warranted, and that he will not be suspected of a mere blunder in syntax or in punctuation.

Rules 3, 4, 5, and 6 cover the most important principles in the punctuation of ordinary sentences; they should be so thoroughly mastered that their application becomes second nature.

In plain terms

Do not put a period in the middle of a sentence and leave part of it standing alone. A phrase or clause that cannot stand by itself belongs with the sentence it depends on, joined with a comma or with no mark at all. Splitting it off creates a piece that looks like a sentence but is not one, which can make writing feel choppy or careless.

BeforeI finally signed up for the pottery class. Because my friend wanted a partner.

AfterI finally signed up for the pottery class because my friend wanted a partner.

Today: Today this mistake is called a sentence fragment, and most guides still advise against fragments in formal writing. Advertising, fiction, and casual writing use them on purpose for emphasis, and Strunk himself allowed an emphatic word to stand alone, but the effect works only when it is clearly a choice.