When Lukashenko came to power, he took a course of rapprochement with Russia. He returned the coat of arms and flag of the Byelorussian SSR instead of ancient Belarusian. People, after the poor start of the 90s, longing for the Soviet Union. He restores the economic and political relations with Russia. The entire industry the last 20 years worked in this direction.

Trends about some social-economic and political problems (results of IISEPS national polls, %)

In part it stems from the fact that the concept of a separate Belarusian ethnicity/nationality only really dates back to around 1890-1900. Prior to that Belarusians were usually either considered Russian or Polish, or even Lithuanian.

During the Soviet Union, the concept of a Belarusian identity was also not widely used. Part of that stems from the Soviets trying to create an all-Soviet identity, but also they held onto the belief that the Russians, Ukrainians and Belarusians were all just subgroups of a larger Slavic nationality; note that the name Belarus comes from Byelorussia, the term used during the Soviet era, and itself simply meaning "White Russia" (the origins of this phrase, which date back centuries, are not fully known).

So when Belarus became independent in 1990/1991, it still had a lot of close ties to Russia. The rise to power of Lukashenko in 1994 also contributed to that. Because it falls within the 20-year rule, I'll be quite brief: Lukashenko is the main driver for bringing Belarus and Russia closer, and for that I can't accurately say why beyond possible economic and cultural reasons, and Lukashenko's apparent fondness for the Soviet Union (note that he was the only deputy in the Belarusian Soviet to vote against independence in 1990). He may have more personal reasons for it, but I'm not to speculate on that.

FYI this probably falls outside the twenty year rule ("history" is defined on this subreddit as being things that happened 20 or more years ago)