Here is one way in which this is true:

One of the major ideas of the Renaissance was the idea that truth comes from empirical proof, not from what authorities have said.  This was the time, for example, when the idea of a geocentric universe with perfectly spherical orbits for everything (which was believed partly on the authority of Aristotle) was being challenged by people based on empirical observation.

In a similar way, Luther challenged the idea that religious truth came from authority.  Instead of letting the Church tell everyone what to believe, he wanted a decentralized system with everyone reading the Bible and figuring out for themselves what God wanted.

How is the Reformation the religious counterpart of the Renaissance?

Seems true to me.  The Renaissance was a time of rebirth and growth in all manner of areas--the arts, science, exploration.  The Reformation was, as its name suggests, a time of change for the church. 

Generally speaking, there is only a need for rebirth or reform when things have been bad.  To put it bluntly, the country and the Church were in trouble. The Catholic Church was corrupt and in need of change, and the country had been at somewhat of a standstill, particularly compared to the glory days of expansion and growth in all areas. 

The Renaissance and the Reformation both unleashed a new and expanding freedom in all things--religious choice, creativity, discovery.

During the Renaissance, there was a breakdown of the political and religious barriers that had tamped down ideas and the exchange of scientific ideas for centuries, coupled with the availability of funding for research, this propelled a large number of new ideas and inventions and all sorts of advancement.

Just so during the reformation, the Catholic Church was shown to have weaknesses and as the various sects began to splinter off and found their own churches and congregations, new interpretations of scripture were suddenly allowed as opposed to being mandated by the Pope or the Catholic Church.  This new openness and the possibility for change and new religious theories created a similar environment in religious thought.