Let's say an LLC owns some shares in company XYZ, and it also owns a small retailing business. The shares are sold and they make a capital gain of $1,000 this year. And the retailing business loses $500 this year. In what ways can the loss reduce the tax burden of the profit?
In no ways.
Both will be reported to the members on their K1 in the respective categories (or if it is a single member LLC - directly to the individual tax return). The capital gains will flow to your personal Schedule D, and the business loss to your personal Schedule C.
On your individual tax return you can deduct up to 3K of capital losses from any other income. Business loss is included in the income if it is active business, for passive businesses (like rental) there are limitations.