No, you are not understanding this correctly. The table shows the required effective key sizes for the different cryptographic primitives.
A cryptographic primitive could for instance be AES, which is a symmetric block cipher. So an AES key should have an effective key size of 128 bits to achieve around 128 bits of security. The best way to attack AES when it is used appropriately is close to brute forcing the key. So the effective size and strength of the AES key is about the same. This is true for any unbroken symmetric cipher: the effective key size is identical to the key strength.
RSA - an asymmetric algorithm - requires a larger key size because the number calculations are on large numbers. For RSA an attacker can try and refactor the modulus to try and find the private key components. It is therefore much easier to attack RSA than to try $2^X$ values for an $X$-bit key. In the table you can see that it will take about $2^{128}$ tests to break a 3072 bit key. So an AES key of 128 bit and a RSA key of 3072 bits both have a strength of 128 bits.
Elliptic Curve cryptography allows for smaller key sizes than RSA to deliver the same strength asymmetric key pair. Generally the effective key size of the key pair needs to be double the size to achieve the same strength as a symmetric key. So we see the value 256 for ECC in that same row. The curve sizes listed are of the named curves first created by Certicom and later standardized by NIST as P-160, P-224, P-256, P-384 and P-521 (that's not a typo, it's not 512).
Note that for RSA the key size is identical to the size of the modulus; encoding all parts of an RSA key will result in a key that is larger than the modulus and thus the key size. The same goes for ECC keys: the size is the size of the order of the curve, it is not the size of an encoded key.
How the keys are used and if you need symmetric or asymmetric (RSA, DH or ECC) depends on your protocol. RSA is not used because RSA keys are larger, it is because RSA keys are asymmetric, which - for instance - allows you to create a public key infrastructure. AES, RSA and DH keys are all used for different purposes.
When creating a protocol or when configuring your software it does make sense to use relating key sizes. So you would use symmetric keys of 128 bit or over, RSA keys of 3072 bits or over and ECC keys (and hashes) of 256 bits or over to achieve an overall strength of 128 bits. This is what you can use the table for.
Using hybrid encryption AES-256 (strength: 256 bit) with an RSA key of 1024 bits (strength 80 bits) doesn't make too much sense; an adversary will try and factor the RSA key because it is much easier to do than to brute force AES. So the strength of the protocol would be limited to 80 bits.
You may be wondering about the 80 bit strength for the symmetric key. There are few symmetric ciphers that use an 80 bit key. There are however known meet-in-the-middle attacks for triple DES. An encoded 128 bit key for (two key) triple DES contains 112 bits that are used within the algorithm but that key only contains about 80 bits of security.
It's recommended to use a three key triple DES key or to move on to AES. Three key triple DES unsurprisingly has a key strength of 112 bits, explaining the second row.
Your table can be found in the second table of the NIST recommendations of 2016 on keylength.com. Keylength.com also contains pointers to the source material (which may be harder to read).
Note that the last Elliptic Curve size in that table reads 512 bits. As mentioned before, NIST however only specifies a 521 bit prime curve, so your table likely references the P-521 curve.
512instead of521$\endgroup$