Introduction of Root key is a critical element in ensuring Break-in recovery. Below is my dig at explaining its necessity -
Why is DH ratchet used:
For argument's sake, if an attacker was able to compromise a given message key, he/she would be able to compute future chain keys and message keys if only Symmetric-key ratchet were used. DH ratchet is hence introduced to prevent this and ensure Break-in recovery.
Now to answer your question:
In DH ratchet, each party generates a new DH key pair and forwards the public key component over to the other party.
Let us go by the assumption that an attacker was able to compromise the chain key at a given point when party A is sending a new DH public key to the party B. Provided that the attacker has the ability to intercept the channel, it would be trivial for him to replace A's DH public key with self generated public DH key and forward the message over to B. In doing so, the attacker can conveniently remain to be an MITM between A and B even with the DH ratchet, unless ---- the DH ratchet were to use an additional secret known only to A and B to derive the chain keys. No points for guessing what the secret is - root key.
Recall that the Root key was bootstrapped at the beginning of establishing the communication between A and B. Since this initialization used pure DH mechanism using Identity keys, there is really no way the attacker could have compromised the root key.
Continuing the narrative - So, when B makes use of attacker introduced public DH key in combination with the root key, B generates a "receiving chain key" and sends along a new public DH key. Our attacker who is intercepting the channel, now attempts to replace B's public DH key with a new self-generated DH public key and sends to A.
The attacker's intention being to to build the corresponding chains to match up to A's sending chain and B' receiving chain to successfully establish a working MITM.
But due to the introduction of 'root key' (or an updated version of it) in the DH ratchet, the attacker will not be able to generate corresponding chain that can quite match up to B's sending chain. The same holds true on the other side with A's receiving chain.
Hence, at this point when B sends a message encrypted using 'message key' derived from B's "sending chain key", the attacker would not be able to make sense of it. Note also that, at this point A and B would have to re-establish trust because the message keys/chain keys don't match up at either ends.
This way, an attacker's window of interception after a chance compromise of a 'message key' is limited until the next opportunistic 'DH ratchet' cycle.
Note that the root key also gets updated with each DH ratchet but there is no way the attacker can influence the root key or glean its contents.