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?

I would recommend to read about a classical one-way ANCOVA model, which potentially (with obvious limitations) fits this study design:

*

*a numeric dependent variable,

*a categorical regressor we are interested in, and

*two numeric regressor that are of no interest to us (covariates).

