I actually don't know why LogMeIn switched from free to charging $100/year to remotely access two computers (or $450/yr to access six or more).
But the reason may be similar to why SugarSync shut off its free Dropbox-wannabe service, as well as other pop-up software firms whose management actually believed the "information wants to be free" hoax. Free doesn't put food on the table (because farmers recognize that food has a cost associated with it) nor does it satisfy investors.
The problem faced by LogMeIn and SugarSync is that consumers don't pay for intangibles. Intangibles are things we cannot touch, such as digital pictures, digitized music, and apps (which are digital). Consumers are keen to greatly overpay for a pair of tangible rip jeans and celebrity-endorsed perfume delivered by trucks, and then compensate by greatly underpaying for vaporous digital products delivered by the invisible Internet.
(Corporations are a different matter; they are willing to pay for security: that the service company will still be around in the future, that they get support and upgrades -- plus, their costs are tax write-offs, unlike for consumers.)
This is why I shudder when Autodesk boasts of 130 million consumer customers, or whatever the number is this week. It is not a number to boast about, because only 1% of the 130 million will ever pay anything, leaving the free-loading 129 million quicksand to Autodesk, who will find itself sinking amidst the cost of servers and electricity, and of employees providing support, programming upgrades, and doing marketing.
Free doesn't work, unless it's your hobby, and even hobbies cost.