Hmmm. Do you have a book or article where this claim is made? It would definitely help to have the context to see what they're getting at.
I can sort of see why this claim might be made if you think of the "interconnected and globalized world" as being "primarily (though not entirely) European states communicating and trading with the most distant corners of their empires." However, I think most historians would see that as problematic for the same reason that claiming colonialism wasn't all bad gets a "Yes, but ... " answer. In short, the native population often saw economic benefit from imperial infrastructure, but that wasn't the reason it was built.
My old economic history professor made this claim as the cornerstone to his discussion of the 19th and 20th centuries. I never really bought it and would love to hear more one way or the other. Was it really as easy to order, say, a bottle of Australian rum, and have it shipped through to the central US?
While modern trade and capital markets are perhaps on par with those of 1870-1914, the costs of migration today (which constitute completely of political barriers) are much higher than those of 1870-1914.
If you are interested in doing some reading about the history of globalization, the WTO's report may interest you (https://www.wto.org/english/res_e/booksp_e/anrep_e/wtr08-2b_e.pdf). When you are reading this, however, keep in mind that the rates they use for comparison will probably paint a different picture of the globalization of the time than using prices of inputs of production (like wages/commodity prices) would. Using prices of inputs of production instead of rates may more accurately reflect market integration. For example, if price gaps are decreasing for a particular good worldwide, this will reflect a decrease in the costs or barriers to trade, which is what economic globalization is all about.
You could make the case that this was true in that time because, in a sense, there were far fewer distinct nations due to the size of the various european empires. For example on this map about 75% of africa is under direct rule of france, britain and belgium. So therefore there are fewer barriers to trade and "more globalised"
I've heard from somewhere that when you consider globalisation to be represented by trade as a percentage of gdp then the world only reached 1914 levels in 1967ish