What does the phrase "adjusted by" mean? 
"Matching of one group with specific exposures or outcome with a
comparable group is frequently used in medical research to investigate
the association between exposure and outcome." Ref

Does it mean that one has to include a factor in a statistical model?
For example, does the sentence:
"The difference in BodyMassIndex under the Factor (adjusted by sex and age) between the control group and treatment group is statistically significant (t value, P < 0.05)."
mean that a researcher considered the following statistical model:
BodyMassIndex ~ Factor + age + sex + error ?
By that, the researcher implicitly assume an equal number of participants in both groups (i.e. balanced design).
In general, the researcher should report BodyMassIndex difference between control and treatment group and mention that the (main) effect of Factor was adjusted by age and sex. In this case, the study design can be unbalanced. In other words, in such a case "adjustment" means including other factors in the model. Correct?
On the other hand, "adjustment" can be done as a "part of the design" and is referred to as "matching". In this case a control group subject and a treatment group subject are of the same sex and the same age. In this case I guess sex and age factors are not included in a model and the study design is balanced. Correct?

"Adjusted by" or "conditioned on" is a phrase that can mean many different things, depending on the context. The New Causal Revolution, led by Judea Pearl and Donald Rubin, provides the best context for understanding what's really going on.
What's really going on is confounding variables. A confounding variable is a variable that sets up a backdoor path from the cause to the effect. Depending on which variables are already being conditioned on, the confounder itself might need to be adjusted for or not. See Pearl's The Book of Why or Causal Inference in Statistics: A Primer to see exactly when you need to adjust for or condition on a variable. Spoiler: a correlated variable is NOT necessarily a confounding variable, and "conditioning on the kitchen sink" can sometimes get you a biased (wrong) answer.
How do you adjust or condition? There are multiple ways to do that:

*

*The first, valid in a regression setting, is simply to include the variable on right-hand side of the model.

*Valid in more general settings, you can stratify your analysis by different values (or ranges of values) of the variable. This works better if you have a sufficient sample size.

*You can use the backdoor adjustment formula (see Pearl's books above) in some circumstances.

*You can use the frontdoor adjustment formula (see Pearl's books again) in some circumstances.

*You can use instrumental variables in some circumstances.

No doubt there are other ways, but these are some of the more common methods.
