Writing simply and concisely is difficult, isn't it?
Both 分かりやすく and 簡潔に are acting adverbially to modify 書く.
Let's start with why you wouldn't use を here. を marks the object of the verb, i.e. it marks the thing that actually gets written, for example, a letter, a poem etc. It doesn't make sense to 'write a brevity' so を is not the correct particle.
簡潔 is a na-adjective. These words can be turned into adverbs by adding に. So 簡潔に書く means 'write concisely'.
やすい is an i-adjective. These words can be turned into adverbs by replacing い with く. So 分かりやすく書く means 'write in a way that is easy to understand'.
If I were to guess に is because its not describing a person
Sorry, but I don't know what you mean by this statement. I can't see any link between using に or を and whether or not you're describing a person. If this is a problem then perhaps you could ask a separate question, giving an example of what you mean.