What do Evangelicals who speak negatively of "religion" mean by that?

Much of the popular usage of "religion" in the negative isn't relegated to any particular denomination of evangelical Protestants, and if you consult theologians in that denomination they wouldn't use it in the negative. Usually, simply quoting James 1:27 gets them saying, "Oh, well not religion like THAT."

When used in the negative, they tend to mean something like, "A system for becoming right with God by your deeds," or "A church or group of Christians preoccupied with rituals, traditions, or doctrinal precision to the expense of a personal relationship with Jesus and loving others."

Evangelicalism is broad. In fact, that is what makes it evangelicalism, as opposed to evangelical Methodism, evangelical Baptist, etc. Evangelicalism is a kind of umbrella category to describe a diverse set of Christians who share a few core features in common concerning Jesus, scripture, and salvation.

Using "religion" in the negative is more of a popular level opinion than a technical term. A common refrain is, "Christianity is not a religion, it is a relationship." I, as an evangelical, disagree with the statement but I get the sentiment.

It's often a loosely used term, much like when a Christian says, "God told me X." If you asked them, "So, when you say that God spoke to you, do you mean that you were thinking and praying about it and you had a really strong impression that you felt comfortable with?" They respond, "Yes, exactly! God just wouldn't let it get off my mind." Rarely would you be told, "No I mean that I heard an audible voice that told me this." Usually, "God spoke to me" is used loosely, and to people with careful distinctions about divine revelation and providence, this manner of speaking is reckless. Yet, it's still a manner of speaking that occurs on a popular level.

I think the same thing is happening with the religion/relationship dichotomy.

Evangelicalism is a pretty wide spectrum, and there are certainly many branches of it that have little issue with calling their belief system a "religion." After all, John Calvin wrote the Institutes of the Christian Religion, and theologians like B. B. Warfield talk about true Christianity as "religion in its purity" (Selected Shorter Writings, I, p. 389).

Many evangelicals, however, particularly those less associated with established denominations, are more likely to accept a "religion vs. relationship" dichotomy, as explained by the evangelical, dispensational, and baptist site GotQuestions.org. In an article entitled, "Is Christianity a religion or a relationship?" they write that in some contexts, the word religion refers to "man-centered" belief systems:

Most religions are similar in that they are built upon the concept that man can reach a higher power or state of being through his own efforts. In most religions, man is the aggressor and the deity is the beneficiary of man’s efforts, sacrifices, or good deeds.

In that regard, Christianity is not a religion; it is a relationship that God has established with His children. In Christianity, God is the aggressor and man is the beneficiary (Romans 8:3). The Bible states clearly that there is nothing man can do to make himself right with God (Isaiah 53:6; 64:6; Romans 3:23; 6:23).

Given their decentralized, non-denominational approach to the church, it's not surprising that they also associate religion with organized religion:

The grace-based relationship between God and man is the foundation of Christianity and the antithesis of religion. Established religion was one of the staunchest opponents of Jesus during His earthly ministry.

Just as the Jewish leaders made a religion out of a relationship with God, many people do the same with Christianity. Entire denominations have followed the way of the Pharisees in creating rules not found in Scripture.

The article concludes by emphasizing the relational aspects of Christianity, which it contends are not found in religion:

[God] does not ask us to try to attain holiness by our own strength, as religion does. He asks that our old self be crucified with Him so that His power can live through us (Galatians 2:20; Romans 6:6). God wants us to know Him, to draw near to Him, to pray to Him, and love Him above everything. That is not religion; that is a relationship.