Likely the term "The Russian Federation" is used to distinguish the Political Entity of the Nation of Russia from the people of Russia and more importantly, the people who are Russian Ethnically.
It's important to note that that throughout history, what was part of "Russia" has changed and included at some point Finland, Poland, Ukraine, Belarus, and other nations. Part of Putin's justification for war is the bring Russian ex-patriots who are living in the Ukraine back under the governernance of the Russian Federation. What's more, the Russian people are an ethnicity of Eastern Slavic peoples. Historically, there are three ethnicities of Russians, though today, the political correctness of this is in dispute. The historical peoples are Greater Russians (Or Red Russians), White Russians, and Little Russians (Or Black Russians). Due to this being in part political propaganda, it's considered more polite by Eastern Slavic people in general to the think of the three ethnicities as Russian, Belarussian, and Ukrainian respectively.
To further confuse matters, there is a large number of ethnic Russians (Red) living in the Ukraine who may not have ever been Citizens of the Russian Federation. And again, while polite, not everyone in the Ukraine consider Ukrainian as an ethnicity and instead seem them the subset of Russian people.
Remember, the war is over who controls regions of Ukraine that culturally identify more with Russia than their own country... and the opinions are not entirely universally accepted in that region either.
So, the reason people in the region will refer to the nation as The Russian Federation is to distinguish the Government of the Nation from the people of the ethnicity who's loyalties both sides want to be sympathetic too as well as the people of Russian Citizenship, who may or may not be Ethnically Russian and many of whom oppose the war. When discussing the actions of the government, simply saying "Russia" is not specific enough (or too likely to be twisted by propaganda into meaning something unintended) while "Russian Federation" specifically refers to the government.
As for why they use the "Russian Federation", it's because that's the official name of the Nation-State commonly called Russia. Almost every nation in the world has an official name that is shortened and in diplomatic circles, it's considered polite to introduce official government representatives as "The [title] of [official name of nation]" or similar (i.e. "The President of the United States of America" vs. The President of America... in fact in the spanish speaking world, especially Latin America, they don't like to refer to people of the U.S.A as "Americans" because that refers to people of both continents... so they refer to Americans as "Estadounidense" which basically means "United Statesians." Likely the only people in Latin America that do not do this are the Mexicans... because this ignores the fact that the official title of their nation is "The United States of Mexico." In fact, "United States" is a possible title used by Federal Nations (Mexico and America are the only two extant ones, but Colombia and Indonesia both used their histories, so calling the people of the U.S.A that is a misnomer along the lines of calling the people of France "Republicans" instead of French because officially France is "The Republic of France".)
So the next logical follow on question is why don't we do the same with Ukraine and refer to it by it's official name rather than just "Ukraine"? Well, that's simple. The official name of Ukraine is "Ukraine". Lots of nations forgo an official name because either they don't care, the name is distinct enough. In the case of Japan, which is officially "Japan", referring to it as The Empire of Japan is considered rude, as it refers to a period in their history from 1865 (the start of the Meiji Period, which the nation rapidly modernized) to 1945 (Specifically the surrender of Japan to the United States that ended WWII) when the Emperor had more political power than in any point prior or since. As they are still have the Emperor, they can't refer to themselves as "The Republic" which means "A nation without a Monarch" and "The Nation of Japan" is too informal.
Most nations tend to be fine with being known by their common name and its acceptable that under less formal conditions, refering to a nation by it's common name is acceptable. In fact, it's a running joke that the more you insist upon using the titles, the more likely those titles are questionable at best. As noted in the show Archer, "The Glorious Democratic People's Republic of North Korea" is none of those things (Rules as written... technically the official title of North Korea is truthful... rules as written, they are a socialist (People's) state that has no monarch (by a bizarre logic that the rightful ruler is a dead guy and his offspring are merely keeping the chair warm for him... no seriously) and they have democratic elections (no one said they had to be a LIBERAL democracy. The difference between a liberal and and an illiberal democracy is largely noted by what happens if you vote in opposition of the current political powers that be). It never claimed to be "glorious", that was just made up by a show writer to sell the joke.
EDIT: TL;DR: The Russia Federation is used to distinguish the speaker is talking about the nation's government, not necessarily the citizens of that nation or the ethnic peoples that share a name with the nation and have a large diaspora from the geographical region and government body, with the vast majority of the dispora living in the Ukraine and their status as Russian citizens and their loyalty to Russia both culturally and politically are one of the central reasons Russia invaded the Ukraine. The Ukraine does not get this treatment because it's official name is "Ukraine" while the official name of Russia is "Russian Federation"
P.S. I'm no Russian speaker or Linguist, but I believe leaving the word "the" out of "Russian Federation" is, I believe, from the fact that Russia has no article words like "The" or "A/An" in English.