"The problem" with consumerism, says a Christianity Today article, is "living to consume." Consumerism "defines our relationships and actions primarily through a matrix of consumption."
In the encyclical, Pope Francis says that consumerism "prioritizes short-term gain and private interest."
Similarly, this blog post by a Lutheran pastor states that consumerism taken to its logical extreme is "being in love with an abundance of 'stuff,' buying goods as a solution for joy, and caring more about material possessions than our neighbor."
The problems with it can probably be most starkly seen when contrasted with its alternatives. The main Christian alternatives to consumerism would be sustainability and charity (in the sense of giving).
Consumerism in contrast to sustainability is the bigger concern for Pope Francis in Laudato si', simply because it's all about care for creation. Consumerist tendencies are not sustainable because natural resources that become the raw materials for consumer products are limited and running out, and in many cases their byproducts pollute the planet. Case in point when Pope Francis identifies consumerism as the "present model of distribution, where a minority believes that it has the right to consume in a way which can never be universalized, since the planet could not even contain the waste products of such consumption."
But charity also comes into play for Francis (unsurprisingly), even in this particular encyclical, for example when he says, "approximately a third of all food produced is discarded, and 'whenever food is thrown out it is as if it were stolen from the table of the poor.'" quoting an address he previously gave in 2013.