Simply because the other company has threatened you with legal action.
Trademark law centers on consumer confusion. If a consumer can mistake your Eego company for the other Eego company (which seems likely), the first company that used the Eego name in commerce can file a trademark infringement suit against the Eego company that came into existence later.
There are several factors that are looked at in trademark infringement suits. How these factors are weighed can show whether there is a likelihood of a lawsuit being filed against you and, if the likelihood is high, whether there is the likelihood of a lawsuit victory for the other Eego company. One factor that will be weighed in determining if a lawsuit makes sense will be if there have been any profits made by your company, or any losses suffered by the other company, that can be attributed to your use of "their" Eego trademark.
If both your company and the other company are operating in the U.S., my feeling is that the other company would not have too difficult a time proving infringement, if the other company has been in operation longer than your company. If the other Eego company federally registered the unique "Eego" name, they will be provided with the most protection offered a company under trademark law. With both Eego companies in the same business (clothing), a trademark infringement suit may become a realistic one to pursue.
You really should have all of the facts of your situation (for example, the dates both Eego's came into being) personally reviewed by an attorney in your area (preferably an IP attorney). Let the attorney see the letter you received from the other Eego company, and let the attorney go over your company's financial picture. Let him compare your business, your logo, your consumer base, your product to the other company. The attorney can advise you on how to respond to the letter after this personal review.
Good luck.