A little bit of terminology first
"evolutionism" refers to a set of beliefs from the 19th century. Evolutionary biology is much more than "evolutionism" and I suppose you are more interested is the current day understanding rather than a historic perspective on the question, is that right?
An "evolutionist" is a person that believes in evolution without necessarily understanding it (as by opposition to "creationist"). Someone who study evolutionary processes is called an evolutionary biologist (or often evolutionary geneticist although it does not exactly refer to the same field).
Issue with the question
It is a very standard question as it is a common fake argument from religious extremists that refuse to accept the theory of evolution. A good answer is necessarily long as (1) there are a number of processes of interest and (2) within the question it is clear that the person asking it has very little understanding in evolutionary biology. I will try to keep as introductory and short as possible. Asking for a short answer for this question is like asking for the name of the most populated city in the USA and authorized to use only one word in the answer (while "New York" is in two words).
Answering to a specific case
For the reasons explained above, it is often easier (faster) to explain for a specific case than in theory for a general case. Here are two examples from Biology.SE
Source of information
There are good books designed for people that know nothing to evolutionary biology to get an introductory answer to that question. The Blind Watchmaker is one such book for example. More generally speaking, one might want to have a look at Understanding Evolution by UC Berkeley, a very introductory source of information on evolutionary biology.
Key Concepts to answer your question
Quickly speaking here are a series of quick concepts that are necessary to understand in order to answer this question.
- Genetic drift
- There are a number of different forces that affect evolutionary processes. Layman often think of natural selection as being the only force causing evolution but this is wrong. Genetic drift refers to the random sampling of individual at one generation to build up the successive generation. The relative importance of drift to natural selection in driving evolution of a population depends on the effective population size. Have a look at Why is the strength of genetic drift inversely proportional to the population size?.
- Typically a phenotypic feature that explain little in the variance of fitness in the population might well increase in frequency in the population just through drift.
- Fitness effect of intermediate states
- It is common for layman to assume that intermediate states ought to be deleterious. This is a typical logical fallacy called argument to ignorance. In reality the intermediate states might well be neutral or only slightly deleterious (and therefore be mainly affect by genetic drift) or might even be beneficial.
- Existence of intermediate states
- Darwin framed his theory into the idea that evolution is a very gradual process. As we know today, this is far from being true. Evolution can actually occur through a big steps. There are single mutations that double the amount of genetic material, there are single mutations that completely affect the physiology or anatomy of an individual. There are cases of horizontal gene transfer (transfer of genetic material between species) as well that can cause sudden important change without having to get through intermediate stages. It is sometimes wrong to assume that there must be intermediate steps.
- Phenotypic flexibility
- While you haven't raised this question, a common creationist argument is to say that if a mutation occurs to make a bone a big longer, then one organism would simultaneously need mutations that would make muscles, nerves and other tissues longer accordingly. In reality our developmental process is quite flexible to an environmental change or to a mutational change. This flexibility is very much used in medicine and agriculture.
Why do we talk about the "Theory" of evolution
This is just a sidenote but you might want to have a look at this post