There are two different kinds of transactions that are implicated by your question.

The first is a stock stock split. This simply turns 1 share into X shares and trades out existing shares for new shares. This has no economic effect and is simply done for the convenience of trading shares for prices that are easy to work work on a practical basis, for example, in software listing stock prices.

The second is an initial public offering, which, confusingly, means not just the first time that newly issued stock is sold to the public by the issuer, but any time that newly issued stock is sold to the public by the issuer.

The possibility of dilution is real in this situation. Abuse of that possibility is primarily managed through the fact that the Board of Directors whose approval is required to issue new stock and sell it to the public, have a fiduciary duty to the company to sell the new stock at a price commensurate with the fair market value of the currently outstanding stock. If the Board of Directors breaches that fiduciary duty to the company, then shareholders may bring what is called a derivative action to enforce the duty that the Board of Directors breached to the company.

Lawsuits of this kind make up a significant share of the total volume of cases in the Delaware Chancery Court, a state court in the U.S. state of Delaware where most large publicly held companies are organized. These cases are also often litigated in the state courts of New York State and of California, both of which are also popular states for publicly held companies to be incorporated in.

Also, keep in mind that the Board of Directors is elected by the existing shareholders, and while most publicly held companies have "Soviet style ballots" in director elections from a pre-set slate of candidates, a proxy fight to run a competing slate of candidates is allowed is existing shareholders are unhappy with their performance. Normally, as a matter of practical reality, the Board of Directors is quite responsive to coalitions of shareholders with substantial holdings who care about the value of their shares as opposed to political issues only tangentially related to the company. A Board of Directors is usually aligned with the interests of the existing shareholders and doesn't have an incentive to screw them over by diluting the value of their shares. People who don't like how a Board is running a publicly held company usually sell their shares and invest in some other company instead, rather than risk being diluted.

SEC approval and approval from state securities regulators is also required for an initial public offering (including a subsequent offering of new shares by a public company) but that approval requires only full disclosure of the transaction and the company's financial situation, and cannot be denied on the basis that the transaction is unfair to existing or to new shareholders on account of the price of the offer.

It is allowed because the whole purpose of a publicly held company is to raise capital to engage in business activities. When a publicly held company needs more money to do something it has several choices:

1. An initial public offering of corporate bonds.
2. An initial public offering of new stock.
3. A private loan from a bank or affluent individual.

Options 1 and 3 increase the debt to equity ratio of the company and increase the company's risk of default on existing loans, while imposing a minimum cashflow requirement that it may not be able to fulfill in the event of a short term economic disruption (like the current pandemic).

In the case of a very big investment, there may simply be no lender who has the capacity to lend enough money privately.

Option 2 makes the company more creditworthy and can be economically desirable to existing shareholders is the shares are sold at a price that makes it a cheaper source of capital than borrowing money at the interest rates currently available to the company. If the capital raised makes possible an investment that increases profits for the company by a greater percentage than the existing shareholder's interests are diluted, it is a good deal for existing shareholders.

"it seems strange that companies would be allowed to raise money by decreasing the value of an asset that shareholders have already bought."

If the new issue is priced right, it should not decrease the value of existing shares. The new stock should be matched by a growth in the assets on the company’s balance sheet. Here’s a simple example:

Suppose you and I own a company with assets worth $100. We each own 5 shares, so each share is worth $10. Suppose we decide to raise capital by selling 5 shares for $50. There are now 5 more shares, but there are also $50 more in assets. Since the company now has $150 in assets, each share is still worth $10.

The more general lesson is this: If the new shares are issued in return for assets, then the new shares won't necessarily reduce the value of existing shares. In fact, if the new capital is used wisely, it should increase the value of the firm by more than the value of the infusion, thus raising the value/share.

What may get reduced is voting power per share. If you and I follow the one share, one vote rule, then we have reduced our voting power. Presumably, existing shareholders realize this, and price the new shares accordingly.

This simple example is a stick-figure rendering of a much more complex reality. But the general idea -- it depends on the balance sheet -- applies in the real world as well.