Manifest means something clear or obvious: something we can see, touch, feel, verify. In terms of education, manifest refers to tangible outcomes: what type of jobs does education lead to, how much money can one make, and what practical skills can be learned that will help make money over the course of a lifetime. For example, a graduate degree from a prestigious university, such as Stanford, might result in gaining a position with a six-figure income. That's a manifest function of education.
Latent means the opposite of manifest: something hidden, dormant, not yet revealed. Because of this, it is more difficult to quantify the latent effects of education. For instance, gaining a college degree does not necessarily result in a high-paying job, nice house, and fancy car. But the process of being educated—of persevering through school—might instill the discipline, desire, and confidence to keep learning and to try to better your financial situation. Also, education builds a store of knowledge that might not apply directly to a job but can be useful in other ways.
For example, a manifest function of a business degree might be a high-level managerial job in a large company. A latent function of the psychology and philosophy courses taken in pursuit of that degree might be the ability to understand employees and motivate them to better performances.