Q - How do we explain the one true Church doctrine to Protestants?
A - Thanks for the question. This can be done in a simple manner or a more complex one. But, before we go further, I would like to make an observation.
We have to be careful in using the label "Protestant". I don't believe this to be your intent, but we have to be careful of putting all Protestants into one neat and tidy box. Sometimes we Catholics seem to think there is just one set of "Protestant" beliefs, but this isn't the case. You can take almost any Christian doctrine, (outside of a few Catholic dogmas - e.g. The Papacy, Marian dogmas, etc.) and find a range of beliefs that different groups of Protestants have on these different doctrines. Thus, before you attempt to explain any doctrine, you should know where the other person is coming from and what they believe personally, because the other thing you need to be careful of avoiding is assuming what the other person knows or believes. While someone may call themselves a Baptist, Methodist, Presbyterian, etc. they may not follow all of the doctrines that a Protestant denomination might teach (this of course goes for individual Catholics as well).
With this in mind, I think that the explanation can be as detailed or as simple as necessary. The things you want to include in any kind of discussion include the following descriptions of apostolic succession and authority:
Christ founded one Church.
The one Church founded by Christ had the Apostles as her leaders after Christ ascended to heaven.
Peter was the leader of the Apostles.
The authority and power of the position of the Apostles was handed down to successors.
The divisions in Christianity today are scandalous.
The Church has both a spiritual and physical reality. Spiritual - all the baptized are members of the Church and are bound together in Christ. Physical - Christ set up a hierarchy with real authority that can still be found in the Bishops of the Catholic and Orthodox Churches.
Something else that should be a part of any discussion on this matter is not that you are trying to "win". Archbishop Fulton Sheen once wrote, "Win an argument. Lose a soul." We should allow the truth of what the Church teaches, by the power of the Holy Spirit, to do the heavy lifting and we should try to get out of the way.