I ain't a lawyer. Just one who studies these things based on my previous work in the domain registration business.
What that phrase meant is there's a chance the domain name you want to register might happen to infringe an existing trademark. You might want to register trendmicrosoftest dot com to eventually sell trendy and very soft pillows, but the domain name contains terms belonging to 2 trademarks (Trend Micro and Microsoft).
One reality about trademark holders is they have to protect their trademark rights for their respective terms and uses or risk losing them. To that end, some employ various means to scour the 'net and see if there's a domain name that's potentially infringing their trademark right.
If they find yours, they might decide to send you a Cease and Desist (C&D) letter demanding you hand it over or else. Some are legit, others aren't.
Trademarks initially arise from use in commerce. By eventually establishing your trademark rights to the term for its specific use (e.g. trendmicrosoftest pillows to sell trendy soft pillows), you are potentially STRENGTHENING your "claim" and preventing others from possibly getting a free ride on it.
If you're US-based, you can file an "intent to use" trademark application at uspto.gov. It'll cost at least $300+, not inclusive of attorney's fees should you retain one to handle it for you.
In essence, establishing a trademark provides you an additional layer of "protection" and possible redress should someone try to "steal" your domain name. It's not a bad idea to demonstrate one, especially if it's for a long-term business.
Two caveats:
1. A domain name can still be "stolen" via technical means (e.g. compromising your login details) even if you have a trademark. A trademark only aims to protect the general public or your market from possibly confusing it with someone else.
2. A trademark is established only for a specific use. It doesn't give you carte blanche over any and all uses of it even if it's unrelated to your specific use.
Getting a trademark might be sensible if you intend to use it for a long-term business. It doesn't completely shield you, but it does give you benefits not ordinarily bestowed on those who don't have any.