Self-assessment is feedback for oneself, that an individual can use for their own learning and improvement.
The ideal context for using self-assessment is on a formative activity, that is focussed on student learning, not task outcomes.
Self-assessment should be anchored to criterias, for example by using a rubric.
Using self-assessment in summative assignments may not yield the same benefits for students, and if used with younger students (high school and below) then the benefits should be made clear to them.
Self-assessment scaffolds student evaluations: students naturally engage in comparative processes, so providing a more scaffolded and guided process for this increases positive outcomes.
Self-assessment encourages self-regulated learning, students examine their learning, and take ownership of it, including reflection, planning and goal-setting.
As Andrade (2019) puts it, self-assessment is feedback, just for oneself . It helps to inform adjustments to processes or products (e.g. essays, presentations) and can aid in enhancing learning outcomes and performance. So the purpose of self-assessment really boils down to generating feedback that an individual can use for their own learning and improvements, in for example their performance.
The ideal context for self-assessment
Although there are many documented benefits of self-assessment for students, the research has been undertaken in many different contexts, so it is also important to understand that these benefits may not be equally likely under all circumstances.
Self-assessment yields the highest benefits for students when assessment is formative and focussed on student learning and not on outcomes. Moreover, self-assessment should be scaffolded, for example by anchoring the self-assessment to defined criteria (such as a rubric) that students understand, this can substantially boost the benefits realized .
When used in summative assessment, students tend to over-evaluate themselves because it is linked to their grade, and their evaluation does often not match that of their instructor . This effect is not seen when used in formative assessment.
A final note is on the cohort you are teaching. Research indicates that university students understand the benefits of self-assessment and therefore are able to reap the benefits, for younger students it may be valuable to make the purpose and outcomes evident to students .
The benefits of self-assessment
Self-assessment can be beneficial for teachers, to help them gain an insight into how students perceive themselves compared to how their peers do, or even whether there was free-riding in group work. But importantly, it is extremely valuable for student learning.
Scaffolds student evaluation : Students naturally engage in comparative processes, where they'll use their own and their peer's work or performance as a frame of reference for comparison. This can be a beneficial process for students to identify areas of improvement, but also their strengths. However, this process can be quite unsystematic, and in that way not so fruitful for students. By implementing a scaffolded self-assessment, it can yield a more guided evaluation and revision, which can make for a more helpful and effective process for students.
Self-assessment linked to self-regulated learning : By encouraging students to examine their learning, they are encouraged to take more ownership of their learning too, including their reflection, self-monitoring, planning and goal-setting. Therefore, encouraging self-assessment also follows the student-centred approach to learning.
Linked to learning and skill development: In general, the research trends indicates that self-assessment is linked to learning and skill development. This can be linked back to the many skills/processes self-assessment activates, such as self-regulated learning, as well as becoming more intellectually humble by gaining an accurate understanding of the limits of your knowledge.
 Andrade, H. L. (2019, August). A critical review of research on student self-assessment. In Frontiers in Education (Vol. 4, p. 87). Frontiers.
 Tejeiro, R. A., Gomez-Vallecillo, J. L., Romero, A. F., Pelegrina, M., Wallace, A., & Emberley, E. (2012). Summative self-assessment in higher education: Implications of its counting towards the final mark. Electronic Journal of Research in Educational Psychology, 10(2), 789-812.
 Race, P. (2001). A Briefing on Self-, Peer, and Group Assessment. LTSN Generic Centre Assessment Series No. 9. York. LTSN Generic Centre.